The exploration of the mapping of Euclidean elements to the Poincare disk is interesting to explore. However, what we will explore here are the mappings of the Poincare disk into the Euclidean plane. We will apply Euclidean definitions within the Poincare disk and observe their mapped effects. next
Starting with this example, let's see what the two lines L and M look like on mapping. next
Figure 6
The mapping of these two lines are arcs of circles in the Euclidean plane. In fact, L and M are Euclidean arcs in the Poincare disk and reflect as such in the Euclidean plane. Let's add a straight line with end points at the edge of the Poincare disk. The mapping of this straight line is itself a straight line which appears to pass through the center of the disk. next
Figure 7
Going back to Euclid's definitions, points and lines (i.e., line segments) are shown reflected here.We note that the mapping of a point is indeed a point. But . . .next
Figure 8
. . . lines J and L reflect as arcs of circles, while line K, which passes through the center of the Poincare circle, reflects as two separate lines. Neither of these two lines touch the circle as they did above. They actually end at a distance from the circle equal to the distance of the respective end points from the circle. go back to image. or next
While, like other circles and rings,these elements beg to be further explored (for some use of their characteristics, refer to the 'Tri as I Might' section of Paul Godfrey's home page), let's continue with other definitions.

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